Strangely, I have had occasion to do a few tutoring sessions with different kids recently around exponential and logarithmic functions.
This particular mistake set off a few alarm bells:
Do you see what the student is doing here? She is treating
like a variable that is being divided instead of a function.
I looked at the student’s notes, and all the usual log laws were there. But she did not yet have the unshakable understanding that logs are functions. I realized that there are some foundational ideas that she needed before we could really make sense of all of this.
Here are a couple of essential ideas I want to communicate to students about logarithm functions.
First, functions can be described as actions, so I always make students explain what a function is doing.
The question you should ask about every function is: what are we doing to the input to get to the output? I call it “saying the function in English.”
Since we usually teach logarithms after exponential functions, let me start with them.
I ask, What do exponential functions do? They provide rules based on repeated multiplication. So the function
tells us that “some number (y) equals 2 multiplied by itself any number of (x) times” to get y. We can do this with different examples, talk about how the function grows, look at the graph, look at tables, compare the growth of exponential functions to linear and quadratic ones. My goal is to get kids to have a feel for what is happening with exponential growth so well that when somebody says, “It was growing exponentially!” they can decide whether that is an accurate statement or not.
This is the first part of the groundwork for understanding logarithms.
Second, remember that anything we do in mathematics, we always find ways to undo.
This is thematic in all of mathematics. It becomes a chant when I teach math.
I say to students:
“Since this is math, anything we learn to do, we need to ….?”
They soon learn to respond with:
Doing-and-undoing is a good mathematical habit of mind to emphasize, because students start to anticipate that when we learn some new funky function or operation, an inverse is coming down the pike. They are not at all surprised to learn that trig functions have an inverse and so on.
In this case, since we have learned to exponentiate, they can guess we need to un-exponentiate.
That’s just how math works!
I like to show inverses of functions in all of the representations. The idea is the same in tables, graphs and equations: the x’s and y’s switch places.
For tables and graphs, it’s fairly easy for students to figure it out. But the algebra gets tricky. To find the inverse of the previous exponential, for example, we need to derive it from:
This immediately creates a mathematical need to “un-exponentiate.”
So when we want to solve that equation for y, let’s undo exponentiation with a function we call a logarithm. Logarithms undo exponentiation.
Since the log undoes the exponentiation, we end up isolating the y.
I also tell them we read this as “log base two of x equals y.”
So when you see an equation like:
you are asking “2 to what power equals 8?” I have them practice explaining what different equations mean.
Now your students are ready to learn all the details of working with logs!
Tell me your ideas in the comments.
[Before I close, vaguely related Arrested Development reference:
Because this is a log law blog. But I guess I don’t really want to talk about log laws. Anyways…]